The New Republic era begins after the Empire's surrender and runs to the eve of the sequel trilogy. In canon it is the home of The Mandalorian and its siblings. In Legends it is the home of the Bantam Spectra boom — the shelf that proved there was an audience for Star Wars fiction at all.
Two shelves, one era
Because of the 2014 continuity reset, this era exists twice over the same thirty years, and the two versions share almost nothing but the calendar. Legends: the New Republic takes Coruscant, Grand Admiral Thrawn nearly takes it back, the Emperor returns in a cloned body, Luke builds a Jedi academy on Yavin 4, and a peace treaty with the Imperial Remnant finally closes the Galactic Civil War in 19 ABY. Canon: the war ends at Jakku in 5 ABY, the New Republic deliberately demilitarises, moves its capital between worlds on principle, and spends thirty years being decent and ineffective while something builds in the Unknown Regions.
Neither replaces the other. This site tags every entry so you always know which shelf you are standing in front of.
Heir to the Empire
Heir to the Empire (Timothy Zahn, Bantam Spectra, 1991) is the book that restarted Star Wars publishing after nearly a decade of silence, and it is the reason this era matters commercially as well as narratively. Read it first, for historical reasons if nothing else. Its antagonist, Grand Admiral Thrawn, is the rare Legends creation important enough that Lucasfilm brought him back into canon rather than leaving him behind — see Reign of the Empire for the canon version.
What the canon era is actually about
Its recurring subject is the cost of principle. The New Republic disarms because a standing fleet is how the last galactic government became an empire. It rotates its capital because a permanent seat of power corrupts. Both decisions are defensible, both are made by people who lived through the alternative, and both are why nobody stops the First Order. Claudia Gray's Bloodline (2016) is the book that makes the argument properly.
Where it ends
In canon, at the Hosnian Cataclysm in 34 ABY, when the First Order destroys the capital system and the Senate with it. In Legends, the era simply flows on into the Yuuzhan Vong invasion — see The New Jedi Order, which is a Legends period and not the canon publishing era of the same name.
If you are new and want one entry point rather than two: read Heir to the Empire for the Legends branch and Bloodline for the canon one, in that order, and you will understand what each continuity thought this era was for.